Harvest and store as much fuel as you can while keeping at least two cargos of it in reserve. If you acquire the technology to harvest stars, you will have no trouble maintaining fuel.
If you can, use technology to scan each planet for the resources it has available. When you have plenty of resources, avoid harvesting planets with poor resources as you may spend more fuel than you gather. Try to limit the fuel spent to 5 or 6 and if you get a lot of resources, expend the same amount the next time. If not, only use 3 or 4 fuel on your next attempt. Always produce more fuel than you waste.
Choose wisely what technology you use. Although the ability to harvest suns is great for fuel, a more useful one allows you to see what kind of planets are in a star system. That way you can target anomalies and breathable planets, while staying away from black holes if you don't have the technology. This allows you to save fuel and know exactly the elements you can collect.
You should drill for materials on inhabited worlds as they also give you the chance to learn about alien technology and refill your oxygen. Previously translated alien keywords will automatically be understood again if encountered later. However, you can manually take a note of what they mean in case you die and lose that ability. This way you can make a more informed decision about a choice that requires alien translation. Note: Starting a new game may randomize the language, so words will be different between playthroughs.
You should switch to larger ships when they become available. However, most new ships you come across will lack important equipment that will make it difficult for you to explore. Transfer all the resources you have on your current ship to the new ship, dismantle all the equipment on your old ship, move it to the new ship, then rebuild the equipment on the new ship.